Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

Syndicate

Validate

September 29, 2010

Jacob Biamonte on Tensor Networks

Posted by John Baez

One of the unexpected pleasures of starting work at the Centre for Quantum Technologies was realizing that the math I learned in loop quantum gravity and category theory can also be useful in quantum computation and condensed matter physics!

In loop quantum gravity I learned a lot about “spin networks”. When I sailed up to the abstract heights of category theory, I discovered that these were a special case of “string diagrams”. And now, going back down to earth, I see they have a special case called “tensor networks”.

Jacob Biamonte is a postdoc who splits his time between Oxford and the CQT, and he’s just finished a paper on tensor networks:

He’s eager to get your comments on this paper, since while it’s aimed at people who already know about tensor networks, it uses a lot of things we talk about here: for example, algebras and coalgebras in symmetric monoidal categories! So, if you folks can’t understand this paper, or find it insufficiently precise, he’d like to hear from you.

Heck, he’d also like to hear from you if you love the paper! But as usual, the most helpful feedback is not just a pat on the back, but a suggestion for how to make things better.

September 23, 2010

Fetishizing p-Values

Posted by Tom Leinster

The first time I understood the problem was when I read this:

PP values are not a substitute for real measures of effect size, and
despite its popularity with researchers and journal editors, testing a
null hypothesis is rarely the appropriate model in science [long list
of references, from 1969 onwards]. In natural populations, the
null hypothesis of zero differentiation is virtually always false, and
if sample size is large enough, this can be demonstrated with any
desired degree of statistical significance.

Bacard on Segal Enriched Categories

Posted by Tom Leinster

The basic aim is
to study categories enriched — in an up-to-homotopy way —
in a bicategory equipped with a class of 2-cells to be thought of as
homotopy equivalences. It’s clear from the introduction that
Bacard’s formalism covers a very wide range of structures indeed. For
instance, you’ll see appearances made by homotopy algebras, torsors,
discrete valuation rings, metric spaces, parallel transport functors,
…. Even motivic cohomology gets a mention.

The exposition looks nice. There are also some unusual and intriguing
pictures. Would anyone like to have a bash at writing a short summary?

September 15, 2010

Grothendieck-Maltsiniotis ∞-categories

Posted by Mike Shulman

Yesterday Georges Maltsiniotis posted a paper on the arXiv in which he presents a definition of ∞\infty-groupoid, said to be due to Grothendieck in Pursuing Stacks, and modifies it to give a similar definition of ∞\infty-categories. (These definitions have been available on his website for a while, but only in French.)

These “Grothendieck-Maltsiniotis” definitions are quite similar to that of Batanin, especially as modified by Leinster. The precise relationship between the two is studied in the thesis of D. Ara, a student of Maltsiniotis, but it’s not hard to get an intuitive idea of their similarities and differences.

September 14, 2010

What Is This Category Enriched In?

Posted by David Corfield

I’ve never felt completely happy with enriched category theory, so perhaps people could help me think through this example.

This question got me wondering again what can be said about a category of conditional probabilities. Let’s take as our objects finite sets, and morphisms to be of the form f:A→Bf: A \to B, a conditional probability distribution Mij=P(bj|ai)M_{i j} = P(b_j|a_i), that is, a row stochastic matrix. This is an arrow in the Kleisli category for the Giry monad. An equivalent category with column stochastic matrices is described by Tobias Fritz in A presentation of the category of stochastic matrices.

September 7, 2010

Grothendieck’s “Tohoku” paper

Posted by John Baez

Michael Barr has spent a lot of time and effort translating Grothendieck’s legendary “Tohoku” paper into English. If you don’t mind violating Grothendieck’s stated wishes, you can now read this translation:

I’m a beginner in integral geometry, which makes this meeting an adventure.
Over the last couple of years I’ve been trying to educate myself in the subject
a bit. In particular, I was in Barcelona for the week before the course, at
the kind invitation of
Joachim Kock. He’s also attending the course, so we spent some time trying to get the basics straight. I think it’s paying off.

Since I’m such a novice, I’m not going to attempt a running summary of the talks. But just to give the flavour of the meeting,
I’ll say a little about each of the first day’s talks.

Posted by Urs Schreiber

Abstract For TT any abelian Lawvere theory, we establish a Quillen adjunction between model category structures on cosimplicial TT-algebras and on simplicial presheaves over duals of TT-algebras, whose left adjoint forms algebras of functions with values in the canonical TT-line object. We find mild general conditions under which this descends to the local model structure that models ∞\infty-stacks over duals of TT-algebras.

For TT the theory of associative algebras this reproduces the situation in Toën’s Champs affine . We consider the case where TT is the theory of smooth algebras: the case of synthetic differential geometry. In particular, we work towards a definition of smooth ∞\infty-vector bundles with flat connection. To that end we analyse the tangent category of the category of smooth algebras and Kock’s simplicial model for synthetic combinatorial differential forms which may be understood as an ∞\infty-categorification of Grothendieck’s de Rham space functor.